Problem: Find the value of $t$ that satisfies $\frac{1}{t+2} + \frac{2t}{t+2} - \frac{3}{t+2} = 3$.
Combining the fractions on the left gives $\dfrac{2t-2}{t+2} = 3$.  Multiplying both sides by $t+2$ gives $2t-2 = 3(t+2)$.  Expanding the right side gives $2t-2 = 3t+6$.  Subtracting $2t$ and 6 from both sides gives $t=\boxed{-8}$.